# What is the smallest prime of the form $2n^n+91$?

I wondered what the smallest prime of the form $2n^n+k$ is for some odd $k$. For $k<91$, there are small primes, but for $k=91$ , the smallest prime (if it exists) must be very large.

• What is the smallest prime of the form $2n^n+91$ ?

It is clear that $\gcd(91,n)=1$ must hold. $2n^n+91$ is composite for every natural number $n$ below $1000$.

$$2\times 15^{15}+91 = 42846499\times 20440124659$$

shows that the smallest prime factor can be large.

• No primes up to $n=1200$. Watch this space... – TonyK May 31 '15 at 11:43
• Did you also check the range $0\le n \le 1000$ ? – Peter May 31 '15 at 11:46
• Yes.${}{}{}{}{}{}$ – TonyK May 31 '15 at 11:47
• No primes up to $n=1500$... – TonyK May 31 '15 at 11:53
• I experience a deja-vu... There must be a glitch in the Matrix ! – Lucian May 31 '15 at 12:35

$$2 \times 1949^{1949} + 91$$
is probably prime! (Running a rigorous primality test on it would take almost a whole day $-$ see here, for instance $-$ so I'm not going to do that.) $2n^n+91$ is composite for all lower values of $n$.